The energy spectrum corresponding to the Coulomb potential

V (r) = ’ 1 is

r

1 1

ElN = ’ (6.69)

2 N + l + D’1 2

2

In (6.69) N and l stand for the radial and angular momentum quan-

tum numbers respectively.

From (6.30) we obtain

1 1

G=’ 2

2 N +l+ D’1

2

1 1

+ 2

2 l + D’1

2

1 1

+ (6.70)

2

2 N ’1+l+j+ D’1

2

© 2001 by Chapman & Hall/CRC

Setting j = 1 it is easy to realize that G becomes independent

of N leading to

1 1

G= (6.71)

2 l + D’1 2

2

Then the general results for H± in the D dimensional space are

1 d2 ±l

El0

H+ ≡ Hl ’ =’ +2

2 dr2 2r

’2

11 1

’+ l + (D ’ 1) (6.72)

r2 2

1 d2 ±l+1

H’ ≡ Hl+1 + G = ’ +

2 dr2 2r2

’2

11 1

’+ l + (D ’ 1) (6.73)

r2 2

where ±l is given by (6.68). For D = 3 these are in agreement with

(6.38) and (6.39).

(b) Isotropic oscillator potential

For the isotropic oscillator potential V (r) = 1 r2 the energy levels

2

are

1

ElN = 2N + l + D , D ≥ 2 (6.74)

2

From (6.30) we ¬nd

D

G=2’ l+j+ (6.75)

2

which turns out to be independent of N . It gives the following

isospectral Hamiltonians

1 d2 ±l

H+ =’ +2

2 dr2 2r

1 D

+ r2 ’ l + (6.76)

2 2

1 d2 ±l+j

H’ =’ +

2 dr2 2r2

1 D

+ r2 ’ l + j + +2 (6.77)

2 2

© 2001 by Chapman & Hall/CRC

These may be compared with (6.43) and (6.44) for D = 3.

To conclude it is worthwhile to note that transformations from

the Coulomb problem to the isotropic oscillator and vice-versa can

be carried out [13-23] and the results turn out to be generalizable to

D-dimensions [13-24].

6.5 References

[1] B.H. Bransden and C.J. Joachain, Introduction to Quantum

Mechanics, ULBS, Longman Group, Essex, UK, 1984.

[2] V.A. Kostelecky and M.M. Nieto, Phys. Rev. Lett., 53, 2285,

1984.

[3] V.A. Kostelecky and M.M. Nieto, Phys. Rev. Lett., 56, 96,

1986.

[4] V.A. Kostelecky and M.M. Nieto, Phys. Rev., A32, 1293, 1985.

[5] R.W. Haymaker and A.R.P. Rau, Am. J. Phys., 54, 928, 1986.

[6] A.R.P. Rau, Phys. Rev. Lett., 56, 95, 1986.

[7] A. Lahiri, P.K. Roy, and B. Bagchi, J. Phys. A. Math. Gen.,

20, 3825, 1987.

[8] J.D. Newmarch and R.H. Golding, Am. J. Phys., 46, 658,

1978.

[9] A. Lahiri, P.K. Roy, and B. Bagchi, Phys. Rev., A38, 6419,

1988.

[10] A. Lahiri, P.K. Roy, and B. Bagchi, Int. J. Theor. Phys., 28,

183, 1989.

[11] A.B. Balantekin, Ann. Phys., 164, 277, 1985.

[12] H. Ui and G. Takeda, Prog. Theor. Phys., 72, 266, 1984.

[13] V.A. Kostelecky, M.M. Nieto, and D.R. Truax, Phys. Rev.,

D32, 2627, 1985.

© 2001 by Chapman & Hall/CRC

[14] B. Baumgartner, H. Grosse, and A. Martin, Nucl. Phys., B254,

528, 1985.

[15] A. Lahiri, P.K. Roy, and B. Bagchi, J. Phys. A: Math. Gen.,

20, 5403, 1987.

[16] J.D. Louck and W.H. Scha¬er, J. Mol. Spectrosc, 4, 285, 1960.

[17] J.D. Louck, J. Mol. Spectrosc, 4, 298, 1960.

[18] J.D. Louck, J. Mol. Spectrosc, 4, 334, 1960.

[19] D. Bergmann and Y. Frishman, J. Math. Phys., 6, 1855, 1965.

[20] D.S. Bateman, C. Boyd, and B. Dutta-Roy, Am. J. Phys., 60,

833, 1992.

[21] P. Pradhan, Am. J. Phys., 63, 664, 1995.

[22] H.A. Mavromatis, Am. J. Phys., 64, 1074, 1996.

[23] A. De, Study of a Class of Potential in Quantum Mechanics,

Dissertation, University of Calcutta, Calcutta, 1997.

[24] A. Chatterjee, Phys. Rep., 186, 249, 1990.

© 2001 by Chapman & Hall/CRC

CHAPTER 7

Supersymmetry in

Nonlinear Systems

7.1 The KdV Equation

One of the oldest known evolution equations is the KdV, named after

its discoverers Korteweg and de Vries [1], which governs the motion

of weakly nonlinear long waves. If δ(x, t) is the elevation of the

water surface above some equilibrium level h and ± is a parameter

characterizing the motion of the medium, then the dynamics of the

¬‚ow can be described by an equation of the form

3 g 2 1

δt = δδx + ±δx + σδxxx (7.1)

2 h 3 3

where the su¬xes denote partial derivatives with respect to the space

(x) and time (t) variables. The parameter σ signi¬es the relationship

3

between the surface tension T of the ¬‚uid and its density ρ : h ’ hT .

3 ρg

It can be easily seen that (7.1) can be put in a more elegant form

ut = 6uux ’ uxxx (7.2)

by a simple transformation of the variable δ and scaling the param-

eters h, ±, and σ appropriately. In the literature (7.2) is customarily

referred to as the KdV equation. Some typical features which (7.2)

exhibit are

© 2001 by Chapman & Hall/CRC

(i) Galilean invariance: The transformations u (x , t ) ’

u(x, t)+ u0 where x ’ x ± u0 t and t ’ t leave the form of (7.2)

6

unchanged.

(ii) PT symmetry: Both u(x, t) and u(’x, ’t) are solutions of

(7.2).

(iii) Solitonic solution: Equation (7.2) possesses a solitary

wave solution

√

a a

u(x, t) = ’ sech2 (x ’ at) , a ∈ R (7.3)

2 2

which happens to be a solitonic solution as well.

Solitary waves in general occur due to a subtle interplay between

the steepening of nonlinear waves and linear dispersive e¬ects [2].

Sometimes solitary waves are also form-preserving, these are then

referred to as solitons. Solitons re¬‚ect particle-like behaviour in that

they proceed almost freely, can collide among themselves very much

like the particles do, and maintain their original shapes and velocities

even after mutual interactions are over.

It was Scott-Russell who ¬rst noted a solitary wave while observ-

ing the motion of a boat. In a classic paper [3], Scott-Russell gives a

fascinating account of his chance meeting with the solitary wave in

the following words

. . . I was observing the motion of a boat which was rapidly drawn

along a narrow channel by a pair of horses, when the boat suddenly

stopped (but) not so the mass of the water in the channel which it had

put in motion; it accumulated around the prow of the vessel in a state

of violent agitation, then suddenly leaving it behind, rolled forward

with great velocity, assuming the form of a large solitary elevation,

a rounded, smooth and well-de¬ned heap of water, which continued

its course along the channel apparently without change of form or

diminution of speed.

We now know that what Scott-Russell saw was actually a solitary

wave. We also believe that the KdV equation provides an analytical

basis to his observations.

© 2001 by Chapman & Hall/CRC

Intense research on the KdV equation began soon after Gardner

and Morikawa [4] found an application in the problem of collision-free

hydromagnetic waves. Subsequently works on modelling of longitu-

dinal waves in one-dimensional lattice of equal mass were reported

[5,6] and numerical computations were carried out [7] to compare

with the recurrences observed in the Fermi-Pasta-Ulam model [8].

The relevance of KdV to describe pressure waves in a liquid gas

bubble chamber was also pointed out [9]. Moreover, the KdV was

found to play an important role in explaining the motion of the

three-dimensional water wave problem [10]. A number of theoretical

achievements were made alongside these. Gardner, Greene, Kruskal,

and Miura [11,12] developed a method of ¬nding an exact solution for

the initial-value problem. Lax [13] proposed an operator approach

towards interpreting nonlinear evolution equations in terms of con-

served quantities, and Zakharov and Shabat [14] formulated what is

called the inverse scattering approach. Further, it was also discov-

ered [15,16] that algebraic connections could be set up by means of

Backlund transformations implying that the solutions of certain evo-

lutionary equations are correlated. We do not intend to go into the

details of the various research directions which took o¬ from these

works but we will focus primarily on the symmetry principles like

conservation laws as well as the solutions of a few nonlinear equa-

tions which can now be analyzed in terms of the concepts of SUSY.

All this will form the subject matter of this chapter.

7.2 Conservation Laws in Nonlinear Systems

The existence of conservation laws leads to integrals of motion. A

conservation law is an equation of the type

Tt + χx = 0 (7.4)

where T is the conserved density and ’χ represents the ¬‚ux of the

¬‚ow. The KdV equation is expressible in the form (7.4), a property it